This phase Ib trial studies the side effects and best dose of ipilimumab when given together with cetuximab and intensity-modulated radiation therapy (IMRT) in treating patients with previously untreated stage III-IVB head and neck cancer. Monoclonal antibodies, such as ipilimumab and cetuximab, may block tumor growth in different ways by targeting certain cells. Specialized radiation therapy, such as IMRT, that delivers a high dose of radiation directly to the tumor may kill more tumor cells and cause less damage to normal tissue. Giving ipilimumab together with cetuximab and IMRT may kill more tumor cells.